1. Principles of Deep Model algorithm Model-Based Approach for...

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Model-Based Approach for Fault Diagnosis.1. Principles of Deep Model algorithm

I-Cheng Chang, Cheng-Ching Yu, and Ching-Tien LiouInd. Eng. Chem. Res., 1994, 33 (6), 1542-1555 • DOI: 10.1021/ie00030a014

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1542 Ind. Eng. Chem. Res. 1994,33, 1542-1555

Model-Based Approach for Fault Diagnosis. 1. Principles of Deep Model Algorithm

I-Cheng Chang, Cheng-Ching Yu,'J and Ching-Tien Liou' Department of Chemical Engineering, National Taiwan Institute of Technology, Taipei, Taiwan 10672, R.O.C.

Equation types of deep models are often employed in fault diagnosis. Upon diagnosis this quantitative process knowledge is utilized as a criterion for satisfaction/violation in a Boolean or non-Boolean manner. Therefore, the resolution of equation-oriented fault diagnosis systems is often limited to, a t most, fault isolation a t a qualitative level. A deep model algorithm (DMA) is proposed to improve diagnostic resolution. First, tolerances of model equations are defined for each model equation with respect to each fault origin. Following the new definition of tolerance, degree of fault is defined to detect the level of fault and a consistency factor is used to evaluate the consistency given by different model equations. A CSTR example is used to illustrate the resolution of DMA. Results show that the proposed method is effective in identifying fault origins.

1. Introduction

Fault diagnosis has received a great deal of attention recently (Willsky, 1976; Himmeblau, 1978; Venkatasubra- manian and Rich, 1979; Mah and Tamhane, 1982; her- mann, 1984; Iri et d., 1985; Kramer and Palowitch, 1987; Ramesh et al., 1988; Davis, 1988; Ulerich and Powers, 1988; Petti et al., 1990; Hoskins et al., 1991; Yu and Lee, 1992; Gertler and Anderson, 1992; Ungar and Psichogios, 1992; Chang et al., 1993). The reason for this is obvious, since chemical plants are operated, if not optimally, a t least safely from an operating point of view. Furthermore, most modern chemical plants are controlled with distributed control systems (DCS). This implies that the availability of process data poses little technical problems for online fault diagnosis. The only problem remaining is an appropriate methodology for utilizing these process data in a real-time environment. Quantrille and Liu (1991) give a good review on process fault diagnosis.

From the rigorousness of the process model employed, the diagnostic systems can be classified into quantitative model-based approach. The Kalman filter approaches (Mah and Tamhane, 1982; Himmeblau, 1978; Fathi et al., 1992), the neutral network approaches (Ungar and Psi- chogios, 1992; Himmeblau, 1989; Hoskins et al., 1991; Fan et al., 1993), expert systems (Dhurjati et al., 1987; Davis et al., 1988), and equation-oriented model-based approaches (Reiter, 1987; Kramer, 1987; Petti et al., 1990; Frank, 1990; Isermann, 1991a and 1991b; Gertler and Anderson, 1992) fall into this category. The signed directed graph (SDG) methods (Kramer and Palowitch, 1987; Chang and Yu, 1990), qualitative simulation (QSIM) approaches (de Kleer and Brown, 1984; Kuipers, 1986; Kramer and Oyeleye, 1988), order-of-magnitude (O[Ml) approach (Mavrovouniotis and Stephanopoulos, 1988), and qualitative process theory (QPT) approaches (Forbus, 1984; Grantham and Ungar, 1991) belong to the qualitative model- based diagnosis. Hybrid systems (semiquantitative model-based) are also proposed (Ulerich and Powers, 1988; Yu and Lee, 1992; Chang et al., 1993).

However, from the resolution point of view, the resolution of a diagnostic system can be up to the quantitative level or the qualitative level. By quantitative level, we mean that the diagnostic system can isolate the fault from the quantitative differences. In other words, if the patterns

* To whom correspondence should be addressed. + E-mail: [email protected].

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of two faults are exactly the same on the basis of the analysis of the signs (e.g., the directions (sign) of the deviations are exactly the same) and differ in magnitude (e.g., the sizes of the deviations are different), then we call the resolution of the system up to the quantitative level. For example, the SDG approach of Kramer and Palowitch (1987) and Chang and Yu (19901, the qualitative physics approach of Kuipers (1986), and the equation-oriented model-based approach of Kramer (1987) and Petti et al. (1990) are diagnostic systems with the resolution up to the qualitative level. On the other hand, the parameter estimation method of Park and Himmeblau (1983), the Kalman filter method of Fathi et al. (1992), and the fault detection, isolation, and accommodation (FDIA) method of Frank (1990), Willsky (1976), and Gertler and Anderson (1992) are the diagnostic systems of the quantitative level.

As far as the knowledge representation is concerned, the model-based diagnostic system can be further classified into equation-oriented (Himmelblau, 1978; Willsky, 1984; Himmelblau et al., 1989; Kramer, 1987; Petti et al., 1990) and graphics-oriented approaches (SDG; Kramer and Palowitch, 1987; Ulerich and Powers, 1988; Chang and Yu, 1990; Yu and Lee, 1992). The advantage of the graphics-oriented approach, e.g., SDG approach, is that the user can visualize the process knowledge. However, at present stage, most process engineers are more familiar with equation-oriented approaches. That is, most process engineers utilize (or were taught) equations to solve engineering problems. Therefore, from the familiarity and maintenance point of view, the equation-oriented approach is an attractive alternative for online fault diagnosis.

The purpose of this series of papers is to investigate and extend the equation-oriented approaches for online fault diagnosis. In part 1, the potential problems for equation- oriented approaches, specifically DMP of Petti et al. (19901, are explored and appropriate remedial actions are also proposed. This paper is organized as follows. Section 2 discusses previous works on model-based diagnosis and illustrates the current problems of equation-oriented diagnostic systems. DMA is proposed in section 3. In section 4, a CSTR example is used to illustrate the consecutive and diagnosis of DMA and a procedure is summarized. A conclusion is given in section 5.

2. Model-Based Diagnosis

Model-based diagnosis systems are fundamentally different from heuristic knowledge-based (rule based) systems

0 1994 American Chemical Society

)Process ’ >

Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994 1543 Faults Disturbances

)Process > input output

1 - - - - ,,,~~ - -

- - - I I

Equations I ’ Equbtion-oriented

1 Diagnosis 1 Syrtam I

input

I

I

output

r - - - - - - - - - - - - i I I

L _ - - - - - _ _ _ _ - A 1, Diagnosis Results

Figure 1. Structure of model-based diagnosis system.

because they rely on the structured knowledge which comes directly from the first principle. Figure 1 illustrates the general concept of model-based diagnosis systems. A fault diagnosis system consists of two major components: a deep model and a reasoning mechanism. In the deep model, the knowledge, either quantitative or semiquantitative, is represented in the form of model equations or in a graphical form (e.g., in the form of SDG). In this stage, any inconsistencies between process model and process measurements are generated. For example, residuals are generated from model equations (Gertler and Anderson, 1992; Petti et al., 1990) or consistencies of specific branches are generated in terms of membership function of the fuzzy set (Yu and Lee, 1992). Typically, the inconsistency measures can be represented in a Boolean (yes or no answer) or non-Boolean (degree of inconsistency) form. The non-Boolean representation receives a great deal of attention. These include the probability assignment (Gertler and Luo, 1989; Gertler and Anderson, 19921, the belief function (Kramer, 1987; Petti et al., 19901, and the fuzzy measure of Yu and Lee (1992). Once an inconsistency is observed, a reasoning mechanism is activated to find the possible fault. In this stage, generally, statistical testing (Gertler and Luo, 1989), evidential reasoning (Dampster- Shafer reasoning, Bogler, 1987; Gertler and Anderson, 1992; Fathi et al., 1993), constraint satisfaction (Kramer, 1987; Petti et al., 1990), and fuzzy reasoning (Yu and Lee, 1992) are employed.

2.1. Equation-Oriented Fault Diagnosis. A fault is understood as any kind of malfunction in the actual dynamic system, the plant, that leads to an unacceptably anomaly in the overall system performance. Such mal- functions may occur either in the sensors, actuators, and process variables or in the components of the process. Fault diagnosis based on static quantitatiue model equations was introduced in the 1970s in the chemical (Himmelblau, 1978) and aerospace (Deckert et al., 1977) industries. Generally, the equation-oriented diagnosis system is formulated in such a way that, initially, the quantitative mathematical model (equations) is abstracted from the first principle. In normal operation, this set of model equations gives zeros for the right-hand-side of the equations’ “parity equation”. In other words, this set of equations is satisfied with normal conditions. Residuals are generated when a process fault occurs. A predeter- mined tolerance is used to indicate possible faulty conditions. Generally, a set of satisfaction factors is generated to give an indication of the violation of model equations (Kramer, 1987; Petti et al., 1990). Following the satisfaction checking, fault discrimination and consistency criterion are applied to isolate the fault. Figure 2 illustrates the general concept of equation-oriented fault diagnosis.

I L------ c i -----A

Diagnosis Results

Figure 2. Structure of equation-oriented fault diagnosis system.

Each component of the diagnostic system is discussed in detail, and resolution and potential problems in this type of diagnostic system are discussed. The development and notations of the system follow the work of DMP (Petti et al., 1990).

2.1.1. Parity Equations. Parity equations constitute the core of the equation-oriented fault diagnosis system. They are generally derived from the material balances and energy balances describing the physical system. Prior to the formulation of the parity equations two sets of parameters have to be specified. One is the “fault” to be diagnosed which is denoted as a vector a = [a l , UZ, ..., a,lT. The fault set a include sensor failure, actuator failure, external disturbances, degradation of equipment, etc. The other is the set of process measurements available which is denoted as m = [ml, m2, ..., mklT. Notice that if the failure of a particular process measurement, e.g., sensor failure, is to be diagnosed, that process measurement is included in the a vector. Following the definition of a and m, a set of parity equations can be expressed as

or, in a more compact form,

c(a, m) = e (2)

Typically, this is a nonlinear set of equations, and at nominal operating conditions the RHS of eq 1 is zero, i.e., el = e2 = ... = e , = 0 for as = [a i , a i , ..., aLIT, where the superscript s denotes the nominal steady-state and a set of linear algebraic equations can be derived:

plla, + plza2 + ... + pli ai + ... + p l n a, + k , = e ,

p j l a, + p j 2 a2 + ... + p j i ai + ... + ~ j , a , + kj = ej

+ prni ai + + Pmn an + k m - - em pml a , + pmz a2 + (3)

or in a matrix form

1544 Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994

P a + k = e (4)

where P is an m X n matrix with the entry pj; and k = [ k ~ , 122, ..., k , ] T . Notice that pi;, kj, and ai are functions of process measurements and system parameters. Here pji can be viewed as the sensitivity of the ith fault (a;) with respect to thejth parity equation at nominal steady-state. Mathematically, this is

(5 )

It should be emphasized that a t nominal operation, the residuals of the parity equations (eq 1 or 3) are zero (el - e2 = ... = e, = 0). That is, when nominal steady-state values are substituted for the fault set (ai = a:) and process measurements (mi = m;), the equations yield zero residuals.

-

P(m')a' + k(m') = 0 (6)

2.1.2. Residual Generation. When a fault occurs (e.g., a; = a: + 6ai), this leads to a new set of process measurements (m). Obviously, a t this point we do not have any knowledge about which ai's constitute the fault (deviate from its nominal value a:). All we can do is substitute as and m into the parity equations. That leads to inconsistency in the parity equations, and residuals are generated.

P(m)as + k(m) = e # 0 (7)

The reason is that a t this new (faulty) steady-state, a consistent set of process variables are m and a*, (a* = as + 6a). That is,

P(m)a* + k(m) = 0 (8)

Without any knowledge about the fault (a* or 6a), residuals are, thus, generated as shown in eq 7. As for the case of sensor failure, ai itself is a process measurement; the explanation is a little different. The correct process measurement, a;, is

a,: = ai,meas + 6Ui

where ai,meas is the measurement reading and 6a; is the bias. Again, the substitution of ai,meas into eq 7 generates residual.

Notice that faulty conditions, e.g., u; = a; + 6ai, are not the only source of the residuals. Measurement noises, modeling error, e.g., P = Pa + 6P, where 6P stands for modeling error, etc. all can contribute to the residuals of the parity equations. In order to achieve robustness in fault diagnosis, some type of checking for satisfaction/ violation in parity equations is necessary.

2.1.3. Satisfaction Factor. Since the residuals arise from not only the fault itself but also from noise effect or modeling error, tolerances (7;s) are used to evaluate the violation of corresponding parity equation cj in a Boolean manner (Venkatasurramanian and Chan, 1989). Kramer (1987) proposes a non-Boolean measure, sf (satisfaction factor), to evaluate the degree of satisfaction to each parity equation. The belief function of Kramer (1987), sf, is defined as

where sgn(e/r) is the sign of e/? which takes the value of

( A ) Pji# 0

I . ,

50 -Tau Tau -1

-50 - 0 Error (B) Pji=O

-1 -50 0 50

Erro r Figure 3. Non-Boolean type of satisfaction factor for (A) pji # 0 and (B) Pji = 0.

+1 when e/? > 0 and becomes -1 when e/? < 0. Therefore, the value of sf falls between -1 and 1 (Figure 3). The non-Boolean type of sf (solid line of Figure 3) avoids abrupt changes from satisfaction to violation of parity equations (compared to Boolean type of sf as shown in Figure 3, dashed line) which, in turn, is more robust to noise effects as well as modeling errors. In the work of Kramer (1987) and Petti et al. (1990), the tolerances (7;s) are defined for each parity equation. Therefore, for a system with parity equations, m tolerances (71, 72, ..., rm) are defined for satisfaction checking. Little emphasis is placed on the selection of 7j's. The selection of ?j's is by no means a trivial matter for fault isolation, as will be discussed in detail later. As pointed out by Kramer (1987) and Petti et al. (1991), the tolerances for the upper bound (77) and lower bound (7;) violations do not have to be symmetric, i.e., 7: # 7L This is helpful for nonlinear chemical processes. The formal definition of sf s is given as follows.

Definition. The vector of satisfaction factor sf = [sfj] = [sfl, sf2, ..., sf,] is the collection of satisfaction factors for each parity equation defined by a belief function (Kramer, 1987) as (a) for high violation

and (b) for low violation

Figure 3 shows the sf for the case of 7: = 7). The shape of the belief function is smooth for one to one mapping to prevent an unstable diagnostic situation.

2.1.4. Fault Isolation. As shown in Figure 2, in the fault isolation stage, an ideal equation-oriented fault diagnosis system consists of two step: fault discrimination (to find the most likely fault) and consistency checking (to check whether the isolated fault is determined consistently from each parity equation). Most of the works

Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994 1545

following set of parity equations:

e, = 2a, + 2a2 + Oa3 - 4

e2 = 2a, - 2a2 + 2a3 - 2

(19)

(20)

(Kramer, 1987; Petti et al., 1990) isolate the fault solely on the basis of the first step, fault discrimination. The consistency checking is considered, a t most, in an indirect manner (Petti et al., 1990).

Consider a set of m parity equations with n faults to be isolated (eq 1 or 3). Once the satisfaction factors, sf, are available (from the previous stage in Figure 2), the fault can be isolated in the following way. The supportability of Kramer (1987) for the ith fault (ai) is defined as

r m - r

where qi is the supportability for the ith fault, r is the number of equations that depends on ai, and (m - r ) is the number of equations that are independent on ai. When qi approaches unity, the result of the reasoning supports the fault assumption. That implies

Pli, ~ 2 i , ~ 3 i , ~ r i + 0 (14)

and

Pr+l,i, ~ r + 2 , i 9 ~ r + 3 , i 9 * * * , P m i = 0 (15)

In other words, the only way to discriminate a fault depends on the zerotnon-zero configuration in the P matrix. Petti et al. (1990) go a step further, using the concept of weighting to improve the diagnostic resolution. The most likely fault is isolated using the failure likelihood (Petti et al., 1990). For the ith fault (ai), the failure likelihood 3i is defined as

PliSf, + PZiSf2 + ... + PmiSfm

lrjlil + B2il + * * e + lrjmil 3i = (16)

where $,i is the normalized parameter with respect to the tolerance 7,, i.e., Pji = p,i/7,. Equation 16 implies that 3i is the normalized weighted sum of sfj’s from all the parity equations. When 3 i approaches +1 (or -11, the reasoning concludes that ai is the most likely fault with a positive (or negative) deviation. Despite the fact that the concept of weighting is more appropriate than the fault isolation using zero/non-zero configuration, it may have difficulties in discriminating the fault even when the faults are qualitatively isolable.

2.2. Resolution. The resolution of equation-oriented diagnostic systems (Kramer, 1987; Petti et al., 1990) is investigated. Let us take a simple set parity equation as an example. Consider a system with two parity equations and three possible faults:

(17)

For the sake of clarity in the illustration, all three faults (a,, a2, and a3) are assumed to be sensor failure. That implies pji’s and Kts are constants in eq 17. In order to evaluate DMP in a consistent manner, the tolerances are defined as

7: = = 7j = min(p. J 1 .lo%,pj2.10%,pj3.10%) (18)

Here ai’s are assumed to be unity for all t ’s . This means that 10% deviation in ai from its nominal value is considered as a fault. Three numerical examples are used to illustrate the resolutions for supportability and failure likelihood.

Example 1. Qualitatively Isolable Faults with Competitive Model Coefficients (pji’s). Consider the

cl(a, m) = ~ , , a , + p12a2 + + t c2(a, m) = ~~~a~ + ~~~a~ + P23a3 + k 2

From the parity equations (eqs 19 and 20), it is obvious that the faults a,, a2, and a3 are isolable in a qualitative manner, since the sign pattern for example 1 is

a1 a2 a3

e1 + + 0 e2 + + (21)

For example, if both eqs 19 and 20 deviate positively (or negatively) then, on the basis of this qualitative observation, we can say that a1 is the fault origin. If el and e2 in the parity equation deviate toward different directions, then a2 is the fault. A similar argument can be applied to the fault a3. Therefore, in our classification the resolution of this system is to the qualitative isolable level. The supportability of Kramer (1987) and the failure likelihood of DMP (Petti et al., 1990) are tested on this example. Table 1A shows the diagnostic results using supportability and failure likelihood with 20 % positive deviation in the fault origins a,, a2, and a3, respectively. Results show that the supportability (qi’s) is not able to distinguished the faults a1 and a2 even when the sign patterns (eq 21) of these two faults are different. The supportability can, however, isolate the fault a3 since the zero, non-zero configuration for a3 is different from that of a1 or a2. The failure likelihood of DMP, on the other hand, cannot distinguish the faults between a1 and a3 (or a2 and a3) when the failure occurs in a1 (or a2) as shown in Table 1A. This simple example shows that despite the fault that quantitative parity equations are employed in the fault diagnosis, the fault isolation approaches using supportability and/or failure likelihood are not able to isolate the fault even when the fault can be isolated from a purely qualitative argument, e.g., eq 21.

Example 2. Qualitatively Isolable Faults with Drastically Different Model Coefficients.

(22)

(23)

-

e, = la, + loa, + Oa, - 11

e2 = loa, - la, + loa, - 19

The sign pattern of this system is exactly the same as that of example 1 (eq 21). That is, the faults a,, a2, and a3 can be isolated simply on the basis of qualitative observation of el and e2. Again, the diagnostic results for 20% positive deviation in a,, a2, and a3, respectively, are shown in Table 1B. A similar interpretation of faults is found using the supportability. That is, the supportability is exactly the same for the faults a1 and a2. The ability to discriminate faults using failure likelihood deteriorates for example 2 as shown in Table 1B. For example, when the fault a1 occurs, the most likely fault interpreted by 3i’s is a3. That means the ability of the approach of failure likelihood to isolate faults for systems with different orders of magnitude in the coefficients decreases. More importantly, it produces erroneous interpretation, e.g., fault origin al or a2 in Table 1B. It should be emphasized that the faults in example 2 can be isolated simply on the basis of qualitative observation.

Example 3. Qualitatively Nonisolable System. Consider another system with two parity equations and three faults:

e, = la, + loa, + loa, - 21 (24)


Table 1. Resolutions for Different Model-Based Diagnostic Algorithms errors statisf. sup p o rta b i 1 it y DMP likelihood this approach

fault origin el e2 sf1 sfz q1 qz 43 F1 Fz F3 ( d h (dz)cn ( d a h

Fault in a1

x. 0 3

0 10 20 10 40 50

'7 error- in a 1

I O

- F7 '?A

0 5 2.

0 0 ' 0 .0 20 30 40 50 r er ror in a1

(A) el = 2al+ 2az + Oa3 - 4 0.4 0.94 0.94 0.88 0.88 0.5

-0.4 0.94 -0.94 0.88 0.88 0.05 0.4 0 0.94 0 0 0.94

(B) el = la1 + lOaz + Oa3 - 11 2 0.94 0.99 0.94 0.94 0.05

2 0 -0.99 0 0 0.99 (C) el = la1 + loa2 + loa3 - 21

0.2 0.94 0.94 0.88 0.88 0.88 2.2 1 1 1.0 1.0 1.0 2 1 0.94 0.94 0.94 0.94

-0.2 0.99 0.94 0.94 0.94 0

Fault in a2 Fault in a3 1 0

~

051 i 0 7 1 ' 1 05:: 3 F I 1 0 0

a 0 10 20 30 40 50

0 0

n 0 I0 20 30 40 50 Cr error in a2 e r ror in a3

0 10 20 30 40 50 0 10 20 30 40 50 0 error in a 3 n

's er ror in a2

31 0 0 1 L& 0 0 0 10 20 30 40 50 0 IO 20 30 40 50 0 10 20 30 40 50 Z error in a 1 7, e r ro r in a2 7 error in a:3

Figure 4. Diagnosis results using supportability (pi) and likelihood (Fi) for example 3.

e2 = la , + l l a , + la, - 13 (25)

The sign pattern for this example is

This means that the propagations of the fault based simply on qualitative observations are exactly the same. Obvi- ously, we do not accept that the approaches of supportability and failure likelihood can discriminate the faults in this example. The results (Table 1C) show that all three faults are indistinguishable for 20% deviation in the fault origin ai. Figure 4 shows the diagnosis results using qi's and 9;s for a range of deviations in al, a2, and a3, respectively. As expected, the supportability and failure likelihood are not able to discriminate the fault for a range of faults.

All three examples show the resolution problems associated with the supportability or failure likelihood. More importantly, the knowledge level of the diagnostic system is up to the quantitative level. Therefore, some modifica- tions have to be made to improve the resolution in the equation-oriented approach.

3. Deep Model Algorithm (DMA)

An algorithm for fault diagnosis, the deep model algorithm (DMA), is proposed to overcome the problems of diagnostic resolution associated with an equation- oriented diagnosis system.

3.1. The Structure. The structure of DMA is shown in Figure 2. The core of the knowledge base for DMA

e2 = 2al-2a2+ 2a3- 2 0.94 0 0.94 (0.94)1 0 0.94 -0.94 (0)-1 0.47 -0.5 0.94 (0.47)o

e2 = loa1 - la2 + loa3 - 19 0.82 0.95 0.99 (0.94)1 0.94 0.82 0.94 (0.5)o 0.9 0.09 0.99 (0.47)o e2 = la1 + llaz + la3 - 13 0.94 0.94 0.94 (0.94)1 0.99 0.99 0.99 (1)1 0.97 0.97 0.97 (0.97)o.w

consists of a set of parity equations (e.g., eq 1 or 3). The parity equations can be either linear or nonlinear. As the process measurements become available, the residuals (ej's) are generated and satisfaction factors (sf s) are computed. The fault can be isolated on the basis of the fault discrimination and consistency checking once sf s are available (Figure 2).

3.2. Tolerance. Despite the fact that most works in equation-oriented diagnosis systems define the tolerance (threshold) for each individual equation (Gertler and Anderson, 1992; Kramer, 1987; Petti et al., 19901, part of the problem in diagnostic resolution comes from the threshold selection; it was clearly shown in example 2. For example, in an equation with imbalance coefficients, the selected tolerance can be too large for one fault and too small for the other. Therefore, a fundamental approach to improve the diagnostic resolution is to define the tolerance on individual equation as well as individual fault basis. For a system with m parity equations and n faults to be diagnosed, m X n tolerance rji (i = 1, 2, ..., m and i = 1, 2, ..., n) is defined. For example, for a fault origin ai, the upper bound violation (1 + &)a: and a lower bound violation (1 - 'yi)aq are the recognized fault, and the upper and lower bound tolerances for the j th parity equation become

(a) violation high

r: = cj(a;, a;, ..., (1 + &)a:, a;+l, ..., a:, m) - 0 (27)

(b) violation low

rL = o - cj(at, ai, ..., (1 - ai)a!, ..., a i , m) (28)

If the linearized version of the parity equations (eq 3) is employed, then the tolerance becomes

- 11

(a) violation high

(b) violation low

(30) rji L = a,pjiaq -

Since the linearized parity equations are used and the upper and lower bound violations are assumed to be the same for clarity, we have

Nonetheless, it should be clear that, in this work, the tolerance (rji) is defined for each fault in each parity equation.

3.3. Satisfaction Factor. With the tolerance (sjj) defined, the satisfaction factor (sfjj) can he calculated once theresidualineachparityequation (ej) isgenerated.Unlike the previous approaches, DMA finds a vector of satisfaction factors for a fault origin. For example, for the fault ai. the vector of satisfaction factor is

T sf, = [sf,, sfze sfse ..., Sfmil

Since the zero/non-zero configuration can be identified by a Kramer approach (Kramer, 1987), a distinction is made between the system with zero and non-zero coefficients. The satisfaction factor is defined for zero @ji = 0 ) coefficients. A new belief function is also defined for the sf with zero coefficient.

The belieffunction for the faultwithnon-zerocoefficient is defined as follows:

(a) for pji # 0

where sfjj is the satisfaction factor of the ith fault on j th parity equation, ej is the residual value of the j th parity equation, rji.is the tolerance of the ith fault on the j th parity equation, and sgn(ejlrji) takes the value of +1,0, or -1 for positive, zero, or negative elements, respectively. Thisdefinitionisexactlythesameas thatofKramer(l987) except that m sfs are computed for a fault. For the case of zero coefficient, the belief function is modified to

(b) for pji = 0

with

T ~ , ~ ~ ~ = min(lrj,l) for all r with pi, # 0 (35)

where 7j,.,,,in is the smallest (in the absolute sense), non- zero tolerance in the j th parity equation and sgn* define the sign of sfji, which can be found from

where SC is the largest sf (in absolute sense) for the ith fault in different parity equations with non-zero coefficients. The reason for going through eqs 35 and 36 to find sfjj (when pji = 0) is that process measurement noises can be misleading to the sign of the sfji when pjj = 0. This is typically true when ai is not the fault origin. Therefore, a positive sign is assigned to sfjj when the largest sf for that particular fault (sfc) does not exceed the threshold (0.1 in eq 36). The belief function with a positive sign is shown in Figure 5. Notice that when sgn* = -1, the belief function is simply the mirror image of that in Figure 5. With this definition, we are able to distinguish the zero/ non-zero configuration.

The characteristicof the proposed method is illustrated in the following example. Consider the set of parity equations in eq 3 with all non-zero pji's. For a given ai, the tolerance (~ji's) can be calculated. That is,


( 'o i i s i s t ( ' i i t 1. Faa.lc~1.

r . f L I

<.I.= I - 1 .

Figure 5. Relationship between ranges of sfj; (sf;- - sf,-) and consistency.

7.. J' = (37)

Now,considerthatfaultaioccurs, withapositivedeviation of ai; the residual in each parity equation is

ej = alpjl.: for j = 1,2, ..., m (38)

With ej's available, we can find the vector of sfi = [sfil, SfZl, ... ( Sf,llT.

.~ ,. . We have sf, = [0.5,0.5, ..., O.5lT. This implies that we are not only able to find the degree of the fault (e.g., 0.5) hut also to measure the consistency for the fault. By consistency we mean that the indication of the fault arises from each parity equation, based on sfi. In theory, the vector of sf s can be utilized to give better diagnostic resolution.

3.4. Fault Isolation. As shown in Figure 2, the fault isolation stage consists of two steps: to find the degree of the fault and to check the consistency of the assumed fault from each parity equation. Two measures are computed for a specific fault, ai: one is the degree of fault di, and the other is the consistency factor cfi once that vector of satisfaction factor, sfi, is available.

3.4.1. Degree of Fault. Since the evidence of fault is generated from the residuals, the method of combination of evidences (e.g., certainty factor in MYCIN, Shortliffe, 1976) is useful in finding the likelihood of the failure ai once the sfjj's are available. Because the residuals are generated from independent parity equations and are further transformed into sfs, a rule for the combinations sfj;s can be helpful in discriminating the fault.

Pettiet al. (1990) utilize the weighted sum of sfstofind the likelihood of failure. This is quite similar to the mechanism of combining evidence in the expert system MYCIN (Shortliffe, 1976; Cendrowskaand Kramer, 1984). Before the degree of fault is defined, it should be emphasized that the sfj(s defined in this work are already weighted by its coefficient @ji) (eqs 29,30,33,34). The degree of fault in ai is

rn

di = ( E f j i ) / m J = 1

di ranges from -1 to 1 and di approaching 1 (or -1) implies that the errors in the parity equations are significant

1548

enough to support the hypothesis, failure in ai with a positive (or negative) deviation. This definition is very similar to the failure likelihood of Petti et al. (1990) except that the weighting is not necessary in this work since sf s are defined differently. Notice that this index di only indicates the degree of the fault (ai), e.g., how close to the threshold the errors are.

3.4.2. Consistency Factor. Since the sfs are defined for the fault in every parity equation, we can utilize this to improve the diagnostic resolution. That is, for a fault ai and a vector of sf, i.e., sfli, sfzi, ..., sfmi, one can use Sfji’S (j = 1, 2, ,.., rn) to further discriminate the fault. A consistency factor cfi is employed to check the consistency in sfji (j = 1,2, .,., rn) generated from the parity equations. Ideally, a complete measure of the likelihood of failure ai is

Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994

where di is the degree of fault for ai and cfi is the consistency factor (similar to the certainty factor in MYCIN, Shortliffe, 1976) for di.

The consistency factor is defined as

where sfmax,i and sfmin,i are the largest and the smallest satisfaction factors, respectively. Sfmax,i and Sfmin,i are defined as

(43)

(44)

Appendix A gives the derivation of cfi. Notice that Cfi ranges from -1 to 1. A positive cfi of unity indicates that the sfji’s = 1, 2, ..., rn) are consistent, and a cfi of -1 reveals that the results from sfji (j = 1, 2, ..., rn) are completely inconsistent. cfi decreases as the consistency between sfji’s decreases. More importantly, qualitatively inconsistent sfji’s (e.g., sfli > 0 and sf2i < 0) lead to a negative value in cfi. Figure 5 shows the characteristic of the consistency factor (cfi). cfi’s are positive for Sfji’S with the same sign, and a Sfma, i > 0 and Sfmin,i = 0 (or Sfm,,i = 0 and sfmin,i < 0) gives a zero cfi. When the range covered by Sfmax,i and Sfmin,min crosses zero, the cfi becomes negative and equal strength Sfji’s (i.e., Isfma,il = IsfmhJ) with opposite sign give cfi = -1, as shown in Figure 5.

With the introduction (diIcfi one can use it to isolate the failure. It should be emphasized that since sf s are defined differently from that of Petti et al. (1990) or Kramer (19871, we are now able to utilize cfi to give a better diagnostic resolution. 3.5. Example Revisited. The resolution of the pro-

posed DMA is examined using the three examples studied. Consider the system in eq 17; the tolerances ~ji’s are defined for each parity equation with respect to different faults. For pji # 0, we have

sfm,,i = max(sfli, sfZi, ..., sfmi)

sfmin,i = min(sfli, sfZi, ..., sfmi)

7ji = Pji.l0% (45)

For the cases with pji = 0, ~ j , ~ i ~ is found using eq 35. With the definition of Tj;lS, Sfji’S can be calculated according to eqs 33 and 34. Once sfji’s are available, one can proceed to calculate di and cfi from eqs 40 and 42, according to the procedure in Figure 2.

Consider the example with qualitatively isolable faults in example 1. The satisfaction factors for a +20% deviation in a1 are

1 0 Fault in a 1

0 0 0 10 20 30 40 50

Z error in a 1

L O r

1- 0 5

0 I O 20 30 40 50 0 0

7 er ror in a1

I O

0 5

0 0 0 IO 20 30 40 50

“3 error in a 1

Faul t in a2

0 5

1 0 Fault in a3

- ” 0 10 20 30 40 SO 9, error in a2

0 0--

- er ror in a 2 0 10 20 30 40 50

0 0 0 10 20 30 40 50 31 er ror in a2

“I

0 10 20 30 40 50

7 e r ro r in a3

I O

d3

0 10 20 30 40 50

? er ror in a3

Figure 6. Diagnosis results using degree of fault (di) and consistency factor (cfi).

The sfji’s indicate inconsistency when a2 (or a3) is the assumed fault (e.g., e1 indicating a positive deviation in a2 and e2 indicating a negative deviation in ad. The degrees of fault (di’s) are 0.94, 0, and 0.47 for al, a2, and a3 with this perturbation (Table 1A). The consistency factors cfi’s also indicate the inconsistency for the fault assumptions a2 and a3 (cf2 = -1 and cf3 = 0) as shown in Table 1A. Clearly, DMA is able to isolate the fault (as opposed to the spurious solutions generated from supportability and failure likelihood). Similar results can also be found for the fault originated from a2 and a3 (Table 1A).

For example 2, an example with qualitatively isolable fault, DMA clearly discriminates the fault using (&fi for all three possible fault origins, as shown in Table 1B. The results from the second example show the advantages of the DMA over the failure likelihood of DMP and the supportability (Table 1B) without incorporating additional process knowledge.

The limitation of the proposed method is illustrated by example 3. Consider the qualitatively indistinguishable example, example 3. That is, the faults give exactly the same pattern qualitatively (e.g., eqs 24 and 25). Obviously, for given fault, all three fault assumptions (al, a2, and a3) are possible fault origins (e.g., spurious solutions) from the supportability and failure likelihood analysis (Table 1C). The DMA also gives spurious solutions, as shown in Table 1C (e.g., fault occurring in a2). Despite the fact that spurious interpretation is minimized to a certain degree (e.g., by comparing the cases for failures in a1 and a3 with these three methods in Table lC), the results clearly indicate the limitation of the proposed method. Figure 6 shows the resolution of DMA for different degrees of faults.

Ind. Eng. Chem. Res., Vol. 33, No. 6,1994 1549

The heat generated from the reaction is removed using a cooling water jacket. Reactor temperature (2') is controlled by changing the set point of a cooling water flow controller (Fjc), and the reactor level (L ) is controlled by changing the outlet flow rate (F). Tuning constants for these two PI controllers are k,l = 32 and 711 = 0.9 h for the temperature loop and kcz = 10 and 712 = 0.6 h for the level loop, respectively. In modeling negligible heat losses, constant densities and perfect cooling water flow control are assumed. Equations describing the system are

Table 2. Steady-State Operating Conditions for CSTR F = 40 ftYh V = 48 ft3

CA = 0.245 mol/ft3 2' = 600 O R

T, = 594.6 "R F, = 49.9 ft3/h VI = 3.85 ft3 ko = 7.08 X 1O"h-' E = 30 000 BTUimol LBet = 0.192 f t k,l = 32 711 = 0.9 h

U = 150 BTU/h ft3 O R

A = 250 ft2

AH = -30 000 BTU/mol cp = 0.75 BTU/lb, O R

cp, = 1.0 BTU/lb, O R

p = 50 lbJft3 p , = 62.3 lb,/ft3 R = 1.987 BTU/mol O R

Ah = 19.6 f t2 Pet = 600 O R

ke2 = 10 112 = 0.6 h

CA, = 0.50 mol/ft3 T,o = 530 O R

Fmax = 96 ft3/h bias2 = 9 psi

bias1 = 12 psi

Table 3. Process Measurements for Fault Diagnosis symbol measured variables

T reactor temp To reactor inlet temp L reactor level

2 reactor outlet flow rate

cooling water outlet temp cooling water flow rate in the jacket

TI

T, 'de 4. Fault Origins svmbol fault orinin

Fo C& ko

U Ti

changes in the feed flow rate changes in the feed concentration changes in the preexponential factor

changes in the overall heat transfer coefficient sensor failure in cooling water outlet temp

of the rate constant

4. DMA and Diagnosis Results A CSTR example (Luyben, 1973; Chang and Yu, 1990;

Yu and Lee, 1992) is used to illustrate the formulation and diagnosis of DMA. Comparisons are made between the DMA and the deep model processor (DMP) of Petti et al. (1990).

4.1. Process Description. An irreversible, exothermic reaction is carried out in a perfectly mixed CSTR, as shown in Figure 7. The reaction is first order in reactant A.

k A-B (47)

- dVT- - FoTo - FT - gkoe-E/RTCAV - -(T UA - Ti) (50) dt P C P P C P

(51) dV.T. UA dt p j c p j

-- ' ' - Fj(Tjo - Tj) + -(T - Tj)

Table 2 gives the steady-state operating conditions. Process variables employed for diagnosis include (1) process measurements and (2) process parameters inferred from control outputs, as shown in Table 3. Faults to be diagnosed include external disturbance (changes in FO or Cb) , equipment degradation (changes in ko or U), and sensor failure (measurement failure in Tj), as shown in Table 4.

4.2. Formulation of Parity Equations. The formulation of parity equations is one of the most important steps in the construction of DMA. For a given system, a straightforward way to formulate the parity equations is to include all governing equations. However, in any realistic situation, not all process variables are measured (or measured online). Therefore, unmeasured variables have to be removed from governing equations.

4.2.1. Elimination of Unmeasured Variables. Fre- quent composition measurements, e.g., measuring CA, is generally not available in an operating environment. Unfortunately, CA plays an important role in the system

sf13 sf14 l:r sfl5

r:; s f ? ' ~ ;;F, sf92 ~ ;;F yft3, , yf?4, ;;I I :f95* 90 -0 5 -0 5 - 3 5 -0 5 -a 5

.I 0 L - i o - I 0 - 1 0 - I 0

Tirne(hr ) Tirne(hr j Time(hr ) Time(hr ) Tirne(hr )

Figure 8. Satisfaction factors (sf,,'s) of CSTR example for fault in FO failure with perfect measurements.


Fault Origin: FO ;'"I FO 2l0i CAO $loT, 1iO :lob U g " ' ~ Tj

0 5 a s 0 5 0 5 0 5

00 :.o 00 00 0 0

-0 5 - 0 5 -0 s

-I 0 - 1 0

-0 5 - 3 i

- 1 0 - - 1 0 - I o

-- I _ 2 3 4 I 7 . 3 . , 2 3 4 3 , 2 3 4 P I 2 3 1

I-- I

Time(hr.1 Tiine(hr ) Time(hr ) Time( hi- ) Time( hr )

, , ?:( h , , "r I I y::i , 0 0

-0 5 -0 5 '0 5 -a 5 -0 5

-I 0 - 1 0 'LO -10 -I 0

Time(hr ) Time(hr ) Time(hr ) Time(hr ) Time(hr ) Figure 9. Diagnostic results of CSTR example for fault in Fo failure with perfect measurements.

Fault Origin FO ( 4 ) I O ,

( B i -::I 'c 3 I s", , q ~ sfp , , 7fY , -0 I -0 s -0 5

- 0 --x - 0 - 1 0 Time(hr ) Time(hr ) Tirne(hr )

Figure 10. Satisfaction factors of DMP approach with (A) perfect measurements and (B) measurement corrupted with noises.

equations (eqs 49 and 50). Therefore, CA has to be eliminated from the parity equation. Since only a static diagnostic system is considered, a simple way to remove CA is to express CA in terms of measured variables. From eq 49, CA becomes

Substituting eq 52 into eq 50 gives the complete set of nonlinear equations describing the CSTR with known variables:

Fo-F=O (53)

-- F VC k e-EJRT

FoTo - F T - A H 0 4 7 0 uA(T - Ti) = 0 (54) PCp (F + koe-E/RTV) PCp

UA Pi%

F,(Tj0 - Tj) + -(T - Tj) = 0 (55)

Obviously, one can utilize eqs 53-55 to generate residuals once the tolerances (eq 5) are available, as mentioned in section 2. For the sake of clarity, in this work, eqs 53-55 are linearized and formulated as a set of linear algebraic equations.

4.2.2. Linearization. The set of nonlinear parity equations are linearized with respect to the fault origins and the known process variables. Appendix B gives the derivation. The resultant linear parity equations are

e, = Fo-F (56)

e2 = 729.68F0 + 40T0 - 871.62T - 699.58F + 1000Tj - 35.7011 + 16OOOCpb - 5.57 X 1O4k0 + 166.67V- 108578

(57)

e3 = 49.90Tj0 + 21.50U + 601.87T - 64.63Fj - 651.78Tj

(58)

4.2.3. Normalization. It can be seen that the coefficients in the parity equations differ by several orders of magnitude. This can lead to problems in online computing. Therefore, a simple way to overcome this is to normalize the fault origins and process measurements such that deviations in these variables are expressed in terms of percent deviation. In other words, the variable ai is normalized with respect to its nominal steady-state values:

Qi Therefore, the linear parity equations become

e, = 40P0 - 40P (60)

e2 = 29187.25p0 + 21182.9T0 - 523002.4T - 27979.5P + 594600Tj + 8000.3P - 5355.930 + 8000.3Cpb +

3946.9h0 - 108578 (61)

e3 = -3225Pj + 26445TjO + 32250 + 361122p- 3875482;. (62)

To simplify these equations, a further normalization with respect to the smallest coefficient of each parity equation is carried out.

e, = Po - P (63)

e2 = 7.395P0 + 5.36713 - 132.51p- 7.089P + 150.65Tj + 2.027P - 1.3570 + 2.027Cb + A, - 27.51 (64)

e3 = -Pj + 8.20pj0 + O S 111.97?'- 120.171;. (65)

where the caron ("1 indicates the normalized variables. In

Q O O Q

-0 5

L?d

; L o o - 0 5 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ -0 5 - 3 5 - 3 5 -0 5 -0 5

_;i , S f t l , 1;‘ Yf’,“ “‘-y-;f ‘,3 * ;4-j;f;4T ;!p,;fz - 1 0 - 1 0 - 1 0 - I 0 - 1 0

Time(hr ) Time(hr ) Time(hr ) Time(hr ) Time(hr )

Figure 11. Diagnostic results of DMP with (A) perfect measurements and (B) measurement corrupted with noises.

Fault Origin: FO

no 0 -0 5 -0 6 -0 5 -0 L -0 s

- L O ’ - I 0 - 1 0 -, 0 - 1 0 Time(hr ) Time(hr ) Time(hr ) Time(hr ) Time(hr )

FO , ““Q g:\i ~ :03 , c, - 21: Tj 0 1 2 3 .

P O 00 ,

-0 5 -0 5 -0 5 -05 ’ 1 2 3 4

,. i, y f ? l , yf?2 * ~ yf?3, s f f 4 , ::j I, yf?5, 00

-0 5 -0 5 -0 5 -0 5 -0 I

- I O - - 1 0 - I O - I 0 - 1 0


-0 5 -0 5 -0 5

L - 1 0 -10 Time(hr ) Time(hr ) Time(hr ) Time(hr ) Time(hr )

Figure 12. Satisfaction factors (sf,,’s) of CSTR example for fault in Fo failure with measurement noises.

a matrix form, the parity equations become

PO 1 0 0 0 0

0 0 0 1.0 -120.17 [ 7.395 2.03 1.0 -1.357 150.65 ] $ +

’j

-132.51T 4- 5.36723 - 7.089F -k 2.03V- 27.51 111.97T- Pj + 8.20Tj0

4.3. Diagnosis Procedure. Following the methodology of DMA, discussed in section 3, the procedure is outlined as follows.

S1. Define the Tolerances for Each Fault Origin with Respect to Each Parity Equation. In the CSTR example, the tolerances are defined as *lo% deviations from nominal steady-state.

0.739 0.2 0.1 -0.135 15.06 (67)

Here rj,,in’s are used for the tolerances with zero coefficients.

52. Generate Residuals. The residuals (ej’s) can be generated in a straightforward manner using eq 66 once process measurements (P, T, To, Tj, Pj, and V) are available. Notice that the fault origins, PO, E&, LO, 0, and Tj, take the values of unity in generating residuals.

53. Calculate the Satisfaction Factors, sfji’s. Once the residuals (el, e 2 , and e 3 ) are available, the sfji’s can be found. Since the zero exists in the parity equations (eq 661, 7 j , m i n ’ ~ are used to find the sf s according to eq 36. The values of 7j,,in’s are defined as the smallest 7j i (in an absolute sense). Therefore, we have 71,min = 0.1 and 73,min = 0.1. Notice that rj,,in is employed for the calculation of sfji by using eqs 33 and 34 for the non-zerolzero configuration.

1 0.1 ‘1,min ‘1,min ‘1,min ‘1,min

73,min ‘3,min ‘3,min 0.1 -12.02


Fault Origin: FO

Time(hr ) Timejhr ) Time(hr ) Timejhr ) Time(hr )

;' ,?I L, ,Ti !$ 41': ,?;I 2 . , 00

-0 6 -0 5 -0 6 -0 8 -0 5

- 1 0 - I 0 - 1 0 - 8 0 - 1 (I

Time(hr ) Time(hr ) Time(hr ) Time(hr ) Time(hr ) Figure 13. Diagnostic results of CSTR example for fault in FO failure with measurement noises.

Fault Origin: CAO


+pv~ -0 5 -0 .Qtq77--? 5 - 0 0 ~ p - - T - r , ~ ~ y o ~ ~ -0 5 -0 0

-, c - I O - L o - - t 0 - 1 0


*o i. -0 5 0 , ,:::io -0 5 :::s;:i -* 5 1 ,::;I -0 5 ~ ,$::I -0 I , ,

-I 0 - 1 0 - 5 0 - I 0 - 1 0


Figure 14. Diagnostic resolutions of DMA for (A) C b failure and (B) ko failure.

54. Calculate Degree of Fault ( 4 ) andconsistency Factor (cfi). Once sfjis are available, di and cfi can be computed according to eqs 40 and 42. The diagnostic results can be interpreted by using the index (diIcfi that di indicates the fault degree and cfi supports the certainty.

4.4. Results and Discussion. Two diagnostic systems, the DMP of Petti et al. (1990) (e.g., the likelihoods 9;'s of eq 13) and the proposed DMA (e.g., fault degrees and consistency factors (di)cfi of eqs 40 and 42) are tested on the CSTR studied by Hsu and Yu (1992). The diagnosis is performed online with a sampling period of 3 min.

Consider the case of a fault being introduced at t = 1 h for a -20 5% decrease in Fo. The sfji's of DMA are shown in Figure 8, and di and cfi of (di),fi are given in Figure 9. The results show that DMA can correctly identify the fault origin FO using (di)cfia Figure 9 reveals that despite the fact that some of the di's are non-zero (even before for

the occurrence of fault, t < 1 h), the combination of di and cfi serves as a useful measure for the fault diagnosis. The DMP, on the other hand, produces spurious solutions as shown in Figures 10A and 11A. It finds Fo, Ch, and Izo are possible fault origins. Notice that, from the SDG-based analyses (Hsu and Yu, 19921, this fault is qualitatively isolable. Unfortunately, the DMP fails to isolate the fault origin. In an operating environment, the measurements are often corrupted with noise. Therefore, the diagnostic systems are tested against process with measurement noises. The temperature (7') and level (L) measurements are corrupted with white measurement noises with vari- ances of 1 O C and 1 5% , respectively. Again, for the fault in Fo, DMA can find the fault origin correctly (Figures 12 and 13) and DMP fails to isolate FO (Figures 10B and 11B).


(A) Fault Origin: CAO

.; il , Li "0' i;i ~ iiq , ill I 'J , coo

-0 5 -0 5 -0 5 -0 5 -0 6

- 1 0 - I 0 - 1 0 - - I O L I O

Time(hr ) Time(hr j Time(hr ) Time(hr ) Time(hr )

(B) Fault Origin kO

CAO

-0 5 -0 5 -0 5

-1 0


Figure 15. Diagnostic resolutions of DMP for (A) C b failure and (B) ko failure.

Fault Origin TJ

Time(hr.) Time(hr ) Time(hr.) Time(hr ) Time(hr.)

- 0 5 t - 1 0 -051 -I 0 I Time(hr ) Time(hr ) Time(hr j Time(hr ) Time(hr )

Figure 16. Diagnostic resolutions of DMA for sensor T, failure.

Next, consider a fault in C b (-20% decrease in Cb). Notice that the SDG-based analyses (Yu and Lee, 1991; Hsu and Yu, 1992) show that the patterns from the fault in C b , ko, and U are not qualitatively distinguishable. However, the equation-based approach (eq 66) shows that C b and ko are not qualitatively isolable (the zero-nonzero configuration and the coefficient are nearly the same for these two faults). Simulation results show that DMA produces spurious solution for this fault. It finds C b and ko as possible faults (Figure 14). Obviously, this result is expected from the analyses of the parity equations (eq 66). The DMP also gives spurious solutions (faults in Fo, C b , and ko, as shown in Figure 15). Similarly, for the situation of sensor Tj failure, the proposed method can correctly find the fault origin, as shown in Figure 16.

Simulation results show that the proposed DMA gives better resolution in fault diagnosis than the DMP. As expected, the resolution of DMA is up to the qualitative isolable level. However, for a given system, the qualitative behaviors from the quantitative parity equations and from the SDG-based model are obviously not the same. The qualitatively indistinguishable faults C b , ko, and U from the SDG model are now reduced to Ch and ko from the parity equations. This can be understood since the quantitative process model is employed in the parity equations and its qualitative behavior depends on the structure of the parity equations.

5. Conclusion

Equation-oriented process models are often used for fault diagnosis. However, the resolution of equation- oriented diagnosis systems is often limited to, at most, the

qualitative level. That is, the extent of quantitative process model is utilized only up to its qualitative level. In order to improve diagnostic resolution, the deep model algorithm (DMA) is proposed for process fault diagnosis using parity equations. The framework of DMA includes a renewed definition of satisfaction factors and the use of di (degree of fault) and cfi (consistency factor) in isolating the fault origin. A procedure is also given for the construction of DMA. A CSTR example is used to illustrate the resolution of DMA. Results show that the proposed DMA is effective in isolating fault origins.

Nomenclature A = heat transfer area of CSTR, ft2 a = vector of fault assumption ai = ith fault c,(.) = j th confluence of system model cp = heat capacity of process liquid, BTU/lb, O R

cpj = heat capacity of cooling water, BTU/lb, "C CA = concentration of reactant A, mol/ft3 Cb = feed concentration of reactant A, mol/ft3 cfi = consistency factor for ith fault di = fault degree of ith fault E = activation energy, BTU/mol e = residual vector e, = residual value of jth parity equation F = reactor outlet flow rate, ft3/h F, = cooling water flow rate, ft3/h Fo = reactor inlet flow rate, ft3/h 3 = failure likelihood vector 3i = failure likelihood of ith fault k = constant vector term of parity equations k, = constant term of jth parity equation

1554

ko = Arrhenius constant, h-l m = vector of process measurements m = number of parity equations n = number of fault origins P = matrix of pJr pll = coefficient for ith fault on jth parity equation q1 = fault supportability of ith fault r = number of equations with non-zero coefficient in parity

sgn* = sign of sfJr, defined by eq 36 sfJ = satisfaction factor of jth confluence, defined by DMP

sfJl = satisfaction factor for ith fault of jth parity equation T = reactor temperature, OR TJ = cooling water temperature, O R

TJo = cooling water inlet temperature, OR To = reactor inlet temperature, OR U = overall heat transfer coefficient, BTU/h ft3 OR V = reactor volume, ft3 VI = heat transfer area of jacket, ft2

Greek letters C Y , = percent of deviation in a, AH = heat of reaction, BTU/mol 7II = reset time for ith loop of PI controller T , ~ = tolerance for ith fault of jth parity equation T, = tolerance for jth confluence of DMP method p = density of process liquid, lbm/ft3 pJ = density of cooling water, lb,/ft3 Subscripts - = lower bound meas = measured data max = maximum value min = minimum value Superscripts - = upper bound " = normalized fault with respect to steady-state value s = steady-state value set = set point

Ind. Eng. Chem. Res., Vol. 33, No. 6, 1994

equations

method

Appendix Appendix A. Derivation of Consistency Factor, cfi.

In a decision-making process, most heuristic methods have sought to justify exhausting information by some qua- siprobabilistic interpretations. The certainty factor CF is the most common representation of heuristic weights that indicates the certainty with which each evidence is believed (Shortliffe, 1976). The value of CF falls between -1 and +1 for the indications of disbelief and belief, respectively. The concept of CF is extended to find the consistency of sfji's for fault isolation.

For a system with m parity equations and a given fault ai, the satisfaction factors sfli, sfzi, ..., sfmi are randomly distributed between the bounds of Sfi,min and Sfi,m=. From the definition of sf;i (eqs 33 and 341, i t is clear that sf;i is the indication of the direction as well as degree of fault. Therefore, sf;i's have to be normalized before it can be processed further. A simple way to do this is to make the largest 1sf;il 0' = 1, 2, ..., m) equal to 1. Therefore, the normalized sf;i becomes (Figure 17)

Sfji si'.. = (Al)

Jr max(lsflil, lsfipl, ..., Isf,,l) Since we have m evidences (sfli, sfzit ..., sfmi) supporting the fault ai, the consistency between these eviqences is a measure of belief or disbelief. For example, if sfji's are all positive and distributed over a small range, then these evidences are consistent (Figure 17A). On the other hand,

Figure 17. Conceptual diagram of consistency factor with decreasing consistency (from A to D).

if sf;i ranges from positive to negative, then these parity equations give conflicting evidence (Figure 17C and DJ. Therefore, the width between the largest and smallest sf;i is an indication of consistency. The consistency factor for the ith fault cfi can be defined as

Cf i = 1 - (sfmmax,i - si'min,i)

Figure 17 shows four cases with decreasing consistency. Appendix B. Linearized Pari ty Equations for

CSTR. The steady-state equations for the CSTR (eqs 53-55) can be linearized with respect to fault origins and process measurements as

O=F,-F (B1)

A A P A ; PIC,;

0 = qT;, = -(F - q) U + -UT - (q - Tj:)F; -

Substituting the steady-state values (Table 2) into eqs Bl-B3, we have

O=F,-F 034)

0 = 729.68F0 + 40T0 - 871.62T - 699.58F + lOOOTj - 35.70U + 16000C4 - 5.57 X k , + 166.67V- 108578

(B5)

0 = 49.90Tj, + 21.50U + 601.87T - 64.63Fj - 651.78Tj

(B6)

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Received fo r review August 19, 1993 Revised manuscript received January 25, 1994’

Abstract published in Advance ACS Abstracts, April 15, 1994.

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